Problem 11 - Calculus: Early Transcendentals

Question

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. \( y = x^2 \) , \( x = y^2 \) ; about \( y = 1 \)

Step-by-Step Solution

Answer

The volume of the solid is zero.

Step 1: Identify the Intersection Points

First, determine the intersection points of the given curves. Set the equations equal to each other: \( y = x^2 \) and \( x = y^2 \). Substituting \( y = x^2 \) into \( x = y^2 \) gives \( x = (x^2)^2 = x^4 \). Solving \( x = x^4 \) gives \( x(x^3 - 1) = 0 \), so \( x = 0 \) or \( x = 1 \). Thus, the intersection points are \((0,0)\) and \((1,1)\).

Step 2: Set Up the Integral for Volume Calculation

We will use the washer method to find the volume. The outer radius at any point \( y \) is the distance from \( y = 1 \) to \( y = x^2 \), which is \( R = 1 - x^2 \). The inner radius is the distance from \( y = 1 \) to \( x = y^2 \), which is \( r = 1 - y \). The limits of integration are from \( y = 0 \) to \( y = 1 \).

Step 3: Compute the Inner and Outer Radii

Calculate the expressions for the outer and inner radii. The outer radius becomes \( R(y) = 1 - x^2 = 1 - y \). The inner radius is a constant \( r(y) = 1 - y \).

Step 4: Formulate the Volume Integral

Formulate the integral using the washer method: \[ V = \pi \int_{0}^{1} [(R(y))^2 - (r(y))^2] \, dy \]Substituting the expressions, \[ V = \pi \int_{0}^{1} [(1 - y)^2 - (1 - y)^2] \, dy \]. As observed, due to symmetry and identical inner and outer radii, the effective volume here simplifies trivially.

Step 5: Evaluate the Integral

Substituting values gives:\[ V = \pi \int_{0}^{1} 0 \, dy = 0 \]Thus, because the outer and inner radii are identical functions along the bounded region, the volume is zero.

Key Concepts

Understanding the Washer Method

The washer method is a technique used in calculus for finding the volume of a solid of revolution. It comes in handy when you rotate a region around a specified line and want to determine the resulting volume. In contrast to the disk method, where there is no inner boundary, the washer method considers both an outer and an inner radius—like a donut with a hole in the middle. Here's how it works:

• The region between two curves is rotated around an axis.
• The solid consists of "washers," or cylindrical slices, with holes.
• The volume of each washer is calculated and summed via integration.

When using the washer method, we need to identify two key components for each washer:

• Outer radius: The distance from the axis of rotation to the outer curve.
• Inner radius: The distance from the axis of rotation to the inner curve.

By determining these radii, the volumetric solution becomes clearer, and you understand how the hollow center affects the solid's volume.

Identifying Curve Intersections

Curve intersections are the points where the given curves meet or cross one another. In our exercise, we have two curves represented as equations: \( y = x^2 \) and \( x = y^2 \). To find where they intersect, we equate them and solve the resulting equation. Setting \( y = x^2 \) equal to \( x = y^2 \), we derive the equation \( x = x^4 \). Solving this equation gives us \( x(x^3 - 1) = 0 \), resulting in intersection points \( x = 0 \) and \( x = 1 \).Corresponding y-values are found by substituting back into either original equation. So, the intersection points stand at \((0,0)\) and \((1,1)\). These points of intersection help in determining the limits of integration, which are essential for setting up the integral accurately.

What Happens During Rotating Regions

Rotating regions involves spinning a flat surface area around a straight line (axis) to form a three-dimensional shape. It's akin to revolving a piece of wire about an axis to create a wheel-like structure. In this exercise, the region enclosed by the curves \( y = x^2 \) and \( x = y^2 \) is rotated around the line \( y = 1 \). This line serves as the axis of rotation, and the solid formed is a symmetrical structure with the upper and lower parts mirroring each other due to symmetry.The rotation of these curves forms a 3D shape where understanding the boundaries' rotation towards the specified axis is crucial. It shows how integration will encompass the entire region above the x-axis to the axis of rotation at \( y = 1 \). This helps visualize how the washer method slices through this revolved area.

Integral Setup for Volume Calculation

The integral setup is the mathematical formation required to compute volumes of the solid of revolution precisely using the washer method. Setting up integrals accurately is crucial since it mathematically models the physical rotation and resulting solid. For our exercise, after identifying the intersection points, we calculate:

• Outer radius \( R(y) = 1 - x^2 = 1 - y \)
• Inner radius \( r(y) = 1 - y \)

These radii help structure the volume integral, resulting in: \[V = \pi \int_{0}^{1} [(R(y))^2 - (r(y))^2] \, dy\] Inserting the expressions leads to the integral: \[V = \pi \int_{0}^{1} [(1 - y)^2 - (1 - y)^2] \, dy\]This particular integral evaluates to zero because the expressions for both radii are identical. Hence, no volume is present, illustrating that sometimes symmetrical rotation nullifies the volume due to identical outer and inner bounds.